ThreeDimensionalConfigurations/DescriptionOfRiemannTypeI

Description of Riemann Type I Ellipsoids

Type I Riemann Ellipsoids

Example Equilibrium Model

This particular set of seven key parameters has been drawn from [EFE] Chapter 7, Table XIII (p. 170). The tabular layout presented here, also appears in a related discussion labeled, Challenges Pt. 2.

$a=a_1=1$

$b=a_2=1.25$

$c=a_3=0.4703$

${\mathrm{\Omega }}_2=0.3639$

${\mathrm{\Omega }}_3=0.6633$

${\zeta }_2=-2.2794$

${\zeta }_3=-1.9637$

As a consequence — see an accompanying discussion (alternatively, ChallengesPt6) for details — the values of other parameters are …

					Example Values
$\tan \theta$	$=$	$-\frac{{\zeta }_2}{{\zeta }_3}\left[\frac{a^2+b^2}{a^2+c^2}\right]\frac{c^2}{b^2}=-0.344793$			$\theta =$	$-19.0238^{\circ }$
$\mathrm{\Lambda }$	$\equiv$	$\left[\frac{a^2}{a^2+b^2}\right]{\zeta }_3\cos \theta -\left[\frac{a^2}{a^2+c^2}\right]{\zeta }_2\sin \theta$			$\mathrm{\Lambda }=$	$-1.332892$
$\frac{y_0}{z_0}$	$=$	$\left[\frac{a^2}{a^2+c^2}\right]\frac{{\zeta }_2}{\mathrm{\Lambda }}=-\frac{b^2\sin \theta }{(c^2{\cos }^2\theta +b^2{\sin }^2\theta )}$			$\frac{y_0}{z_0}=$	$+1.400377$
$\frac{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}{y_{\mathrm{m}\mathrm{a}\mathrm{x}}}$	$=$	$\left\{\mathrm{\Lambda }\right[\frac{a^2+b^2}{b^2}]\frac{\cos \theta }{{\zeta }_3}{\}}^{1/2}$			$\frac{x_{\mathrm{m}\mathrm{a}\mathrm{x}}}{y_{\mathrm{m}\mathrm{a}\mathrm{x}}}=$	$+1.025854$
	$=$	$\frac{(c^2{\cos }^2\theta +b^2{\sin }^2\theta {)}^{1/2}}{bc}$
$\dot{\phi }$	$=$	$\left\{\mathrm{\Lambda }\right[\frac{b^2}{a^2+b^2}]\frac{{\zeta }_3}{\cos \theta }{\}}^{1/2}$			$\dot{\phi }=$	$+1.299300$

See Also

• Description of Riemann Type I Ellipsoids
• Riemann Type 1 Ellipsoids (old introduction)
• Construction Challenges (Pt. 1)
• Construction Challenges (Pt. 2)
• Construction Challenges (Pt. 3)
• Construction Challenges (Pt. 4)
• Construction Challenges (Pt. 5)
• Construction Challenges (Pt. 6)